926 CHAPTER 13 Counting and Probability elements represent all the possibilities that can occur as a result of the experiment. Each element of S is called an outcome .To each outcome a number is assigned, called the probability of that outcome, which has two properties: • The probability assigned to each outcome is nonnegative. • The sum of all the probabilities equals 1. DEFINITION Probability Model A probability model with the sample space … { } = S e e e , , , n 1 2 where … e e e , , , n 1 2 are the possible outcomes and … ( ) ( ) ( ) P e P e P e , , , n 1 2 are the respective probabilities of these outcomes, requires that … ( ) ( ) ( ) ≥ ≥ ≥ P e P e P e 0, 0, , 0 n 1 2 (1) ∑ ( ) ( ) ( ) ( ) = + + + = = P e P e P e P e 1 i n i n 1 1 2 (2) Determining Probability Models In a bag of M&Ms ® , the candies are colored red, green, blue, brown, yellow, and orange. A candy is drawn from the bag and the color is recorded. The sample space of this experiment is { } red, green, blue, brown, yellow, orange . Determine which of the following are probability models. EXAMPLE 2 Outcome Probability red 0.3 green 0.15 blue 0 brown 0.15 yellow 0.2 orange 0.2 (a) Outcome Probability red 0.1 green 0.1 blue 0.1 brown 0.4 yellow 0.2 orange 0.3 (b) Outcome Probability red 0.3 green −0.3 blue 0.2 brown 0.4 yellow 0.2 orange 0.2 (c) Outcome Probability red 0 green 0 blue 0 brown 0 yellow 1 orange 0 (d) Solution (a) This model is a probability model because all the outcomes have probabilities that are nonnegative, and the sum of the probabilities is 1. (b) This model is not a probability model because the sum of the probabilities is not 1. (c) This model is not a probability model because P (green) is less than 0. Remember that all probabilities must be nonnegative. (d) This model is a probability model because all the outcomes have probabilities that are nonnegative, and the sum of the probabilities is 1. Notice that ( ) = P yellow 1, meaning that this outcome will occur with 100% certainty each time that the experiment is repeated.This means that the bag of M&Ms ® contains only yellow candies. Now Work PROBLEM 7
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